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I have a wire that stretches from $x=0$ to $\infty$. The temperature at $x=0$ is given by the unknown function $f(t)$ for $t$ from $-\infty$ to now ($t=0$).

I can measure the temperature of the wire at each point now ($t=0$), $g(x)$.

Given the temperature of the wire now, $g(x)$, I would like to recover $f(t)$, the temperature at $x=0$, through history

The problem is likely "ill posed" meaning tiny errors in $g(x)$ lead to big errors in $f(t)$.

What is known about solving this problem? For example, is the degree of ill-posedness known? If it is mildly ill-posed, are there numerical techniques available? Can anyone point me to articles or a book that treats it?

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You are right, this is a classically ill-posed question, and here is why.

If you are measuring the temperature $k$ along the wire at position $x$ and time $t$, there are any number of different initial temperature distributions along that wire which would yield $t = k$ at $(x,t)$—that is, the answer is not unique for any given set of initial conditions.

This means you can predict what $k = f(x,t)$ will be for future times, but not what it was for past times.

Another way of thinking about this is as follows: heat flux travels by a diffusion process. At any stage of the diffusive process, it tends to erase any historical information about the initial conditions present at earlier times. That erasure makes it mathematically impossible to reconstruct the initial conditions by back-calculation.

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    $\begingroup$ alternatively, one could think of the inverse-time evolution of this equation to become multi-valued, as some kind of superposition (no relationship with algebraic linear superposition, that I'm aware) of well defined previous states $\endgroup$
    – lurscher
    Oct 17 at 22:06
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    $\begingroup$ I believe it is ill posed in a similar sense to inverting the Laplace transform. I suspect that mathematically it has an inverse, but practically/numerically it is hard or impossible. I would like to find out in detail what is known about the maths of this problem $\endgroup$
    – Peter A
    Oct 17 at 22:08
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    $\begingroup$ "the answer is not unique for any given set of initial conditions" I am not convinced this is actually correct. I expect the time-reversed heat equation $\frac{du}{dt} = -\Delta u$ to have a unique solution, at least in some suitable Sobolev space... one can construct such a solution using Fourier series, for instance. $\endgroup$
    – user2617
    Oct 18 at 10:18
  • $\begingroup$ I believe you are right mathematically. But it is numerically ill-posed. Some very helpful answers already, but I am still looking for an article or book specifically covering the heat equation. Eg, proof that it is ill-posed. Degree of ill-posedness. Numerical methods specifically applied to the heat equation $\endgroup$
    – Peter A
    Oct 18 at 10:52
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    $\begingroup$ For a wire stretching to infinity, doesn't uniqueness fails for future times as well? At least without some restrictions on heat distribution. $\endgroup$
    – Wojowu
    Oct 18 at 12:29
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There are indeed some numerical methods available that try to limit the impact of the ill-posedness.

Often times they rely on the introduction of a generalization term. This is a term that in a way measures the errors introduced by the ill-posed nature and adds this to minimization problem of finding your initial state.

This is actually used in a well-known machine learning technique called Support Vector Machines, where you have a similar trade-off between finding a perfect fit for your data (which is very noise-sensitive) and limiting the complexity of the model. In machine learning this is more often identified as the bias-variance tradeoff, but in essence these 2 concepts are very close to eachother.

Since you requested some literature here are some sources that I found usefull during my masters research on these topics:

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